Finite Group Databasebeta

← #13160061

Subgroups — #13160061

Order 1728.

1128 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×Dic6C211yesmaximal, characteristic
2C2×#88853C211yesmaximal, characteristic, complemented
3#11723431C211yesmaximal, characteristic
4C2×#88799C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic
6C32×#816C211yesmaximal, characteristic, complemented
7#11722041C211yesmaximal, characteristic, complemented
8#10592508—31nomaximal
9#10590952—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293—21no—
15C2×Dic3×C3⋊D3—21no—
16C3⋊D3×Dic6—22no—
17C3⋊D3×Dic6—22no—
18C2×C33×C8C2211yescharacteristic
19#88853—22no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22Dic3⋊(C3⋊Dic3)—21no—
23#88803—21no—
24#88799—22no—
25C2×Q8×C3⋊D3—31no—
26C2×D3×Dic6—34no—
27C2×#3756—31no—
28C2×#3746—31no—
29C2×#3746—34no—
30C2×#3746—64no—
31#64106—31no—
32#63857—34no—
33C2×#3719—34no—
34C32×#103—31no—
35#63647D311yescharacteristic
36#63637—31no—
37C3×(C2×D3)⋊C8—34no—
38C3×#816D314yescomplemented
39#63567—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308—21no—
46#6308—21no—
47#6308D412yes—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6D412yes—
55#6293—24no—
56Dic3×C3⋊D3—24no—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#4809—94no—
60#4739—91no—
61C2×C32×Q8—31no—
62Q8×C3⋊D3—62no—
63Q8×C3⋊D3—62no—
64C2×C4×C3⋊D3—61no—
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×#634—31no—
67C2×#634—34no—
68C2×#634—31no—
69C2×#634—34no—
70C2×#634—61no—
71C2×#634—64no—
72C2×#634—64no—
73C2×C3×C4×D3—34no—
74C2×C3×Dic6C2×D314yes—
75C2×#627—64no—
76C2×D3×Dic3—64no—
77D3×Dic6—68no—
78D3×Dic6—68no—
79C2×C32×C8—24no—
80C2×C32×C8C2×D314yes—
81C2×C32×C8C2×D311yescharacteristic
82C32×C4⋊C4—31no—
83#3756—62no—
84#3752—61no—
85C4⋊(C3⋊Dic3)—61no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87#3746—62no—
88#3746—68no—
89#3746—616no—
90C3×C4⋊Dic3C2×D314yes—
91C2×C3×C3⋊C8—34no—
92Dic3⋊Dic3—64no—
93(C2×D3)⋊Dic3—64no—
94#3719—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3Q1612yescomplemented
99C33×C4C2×D411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3⋊(C3⋊Dic3)—22no—
102C3⋊(C3⋊Dic3)—41no—
103C3×C3⋊Dic3—21no—
104C32×Dic3—22no—
105C32×Dic3—41no—
106C2×D3×Q8—94no—
107C22×Dic6—91no—
108C2×#216—94no—
109#924—91no—
110C2×Dic12—91no—
111C2×Dic12—94no—
112C2×Dic12—91no—
113C2×Dic12—184no—
114#897—94no—
115C3×#103—34no—
116C3×C22⋊C8—91no—
117(C2×D3)⋊C8—34no—
118#816—91no—
119#816C3⋊D311yescharacteristic, complemented
120#810—94no—
121C22×C3⋊D3D1211yescharacteristic
122C22×C3×D3—34no—
123C32×Q8—32no—
124C32×Q8—61no—
125C2×C32×C4—24no—
126C2×C32×C4—61no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—61no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—124no—
135C2×C3⋊Dic3—124no—
136C2×C3⋊Dic3D1211yescharacteristic
137C4×C3⋊D3—21no—
138C4×C3⋊D3—21no—
139C4×C3⋊D3—64no—
140#634—32no—
141#634—38no—
142#634—68no—
143#634—64no—
144#634—61no—
145#634—68no—
146#634—64no—
147#634—61no—
148#634—61no—
149#634—64no—
150#634—64no—
151#634—124no—
152#634—124no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—34no—
156C3×C4×D3—64no—
157C3×C4×D3—64no—
158C3×Dic6—24no—
159C3×Dic6D1218yes—
160#627—616no—
161D3×Dic3—616no—
162C32×C8—28no—
163C32×C8—24no—
164C32×C8—21no—
165C3⋊(C3⋊C8)—21no—
166C3×C3⋊C8—64no—
167#451—271noSylow
168C2×C33#12011yescharacteristic
169C2×C33C2×Q1611yescharacteristic
170C2×C33#13711yescharacteristic
171C3×C3⋊D3—22no—
172C2×C3×Q8—34no—
173C22×C3×C4—91no—
174C22×Dic3—181no—
175D3×Q8—188no—
176D3×Q8—188no—
177C2×C4×D3—34no—
178C2×C4×D3—184no—
179C2×Dic6—94no—
180C2×Dic6—91no—
181C2×Dic6—94no—
182C2×Dic6—91no—
183C2×Dic6—184no—
184C2×Dic6—184no—
185C2×Dic6—182no—
186C2×Dic6—181no—
187C2×Dic6—184no—
188C2×Dic6—182no—
189C2×Dic6C2×C3⋊D311yescharacteristic
190C2×C3×C8—24no—
191C2×C3×C8—91no—
192C2×C3×C8D3214yes—
193C2×C3×C8C2×C3⋊D311yescharacteristic
194C3×C4⋊C4—34no—
195#217—181no—
196#216—188no—
197(C2×D3)⋊C4—184no—
198C4⋊Dic3—184no—
199C4⋊Dic3C2×C3⋊D311yescharacteristic
200Dic3⋊C4—181no—
201C2×C3⋊C8—34no—
202Dic12—188no—
203Dic12—182no—
204Dic12—1816no—
205Dic12—182no—
206C22×C32—24no—
207C22×C32C2×D1211yescharacteristic
208C22×C32D3×D414yes—
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3Dic1212yescomplemented
212C2×C3×D3—38no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—62no—
218C32×C4—121no—
219C32×C4C2×D1214yes—
220C32×C4D3×D414yes—
221C32×C4D3×D411yescharacteristic
222C32×C4D3×D411yescharacteristic
223C3⋊Dic3—21no—
224C3⋊Dic3—62no—
225C3⋊Dic3—68no—
226C3⋊Dic3—128no—
227C3⋊Dic3—128no—
228C3⋊Dic3—124no—
229C3⋊Dic3—121no—
230C3×Dic3—28no—
231C3×Dic3—44no—
232C3×Dic3—64no—
233C22×Q8—271no—
234C2×Q16—271no—
235C2×Q16—271no—
236C22⋊Q8—271no—
237#103—31no—
238#103—271no—
239C22⋊C8—91no—
240C33#45111yescharacteristic, complemented, Sylow
241C23×C3—91no—
242C22×D3—34no—
243C3×Q8—38no—
244C3×Q8—64no—
245C2×C3×C4—24no—
246C2×C3×C4—64no—
247C2×C3×C4—64no—
248C2×C3×C4—91no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—34no—
256C2×Dic3—184no—
257C2×Dic3—184no—
258C2×Dic3—181no—
259C2×Dic3—184no—
260C2×Dic3—181no—
261C2×Dic3—364no—
262C2×Dic3—364no—
263C4×D3—64no—
264C4×D3—64no—
265C4×D3—1816no—
266Dic6—21no—
267Dic6—98no—
268Dic6—92no—
269Dic6—188no—
270Dic6—184no—
271Dic6—1816no—
272Dic6—184no—
273Dic6—181no—
274Dic6—188no—
275Dic6—184no—
276Dic6—184no—
277Dic6—181no—
278Dic6—181no—
279Dic6—364no—
280Dic6—364no—
281Dic6C4⋊(C3⋊D3)12yes—
282C3×C8—28no—
283C3×C8—24no—
284C3×C8—21no—
285C3×C8—181no—
286C3⋊C8—64no—
287C2×C32—24no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32#88214yes—
291C2×C32#88211yescharacteristic
292C2×C32C2×Dic1211yescharacteristic
293C2×C32#90911yescharacteristic
294C2×C32#91514yes—
295C2×C32D3×Q1614yes—
296C3⋊D3—22no—
297C3×D3—68no—
298C2×Q8—31no—
299C2×Q8—271no—
300C2×Q8—271no—
301C2×Q8—541no—
302C2×Q8—542no—
303C2×Q8—542no—
304C22×C4—91no—
305C22×C4—541no—
306Q16—542no—
307Q16—542no—
308C2×C8—91no—
309C2×C8D3×C3⋊D311yescharacteristic
310C4⋊C4—31no—
311C4⋊C4—541no—
312C22⋊C4—541no—
313C22×C3—24no—
314C22×C3—92no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3D3×D1214yes—
318C22×C3D4×C3⋊D311yescharacteristic
319C2×D3—38no—
320C2×D3—64no—
321C2×D3—64no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—68no—
325C3×C4—124no—
326C3×C4—181no—
327C3×C4D3×D1214yes—
328C3×C4#381114yes—
329C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
330C3×C4D4×C3⋊D311yescharacteristic
331Dic3—22no—
332Dic3—41no—
333Dic3—64no—
334Dic3—182no—
335Dic3—188no—
336Dic3—368no—
337Dic3—368no—
338Dic3—361no—
339Dic3—364no—
340C32—24no—
341C32#473911yescharacteristic, complemented
342C32#480914yescomplemented
343C23—91no—
344Q8—32no—
345Q8—61no—
346Q8—272no—
347Q8—544no—
348Q8—541no—
349Q8—541no—
350Q8—541no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—91no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—544no—
357C2×C4—541no—
358C2×C4—541no—
359C2×C4C2×D3×C3⋊D311yescharacteristic
360C8—21no—
361C8—181no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—182no—
366C2×C3D3×Dic1214yes—
367C2×C3#6380014yes—
368C2×C3#6392314yes—
369C2×C3#6409111yescharacteristic
370C2×C3#6412411yescharacteristic
371C2×C3Q16×C3⋊D311yescharacteristic
372D3—68no—
373C22—92no—
374C22—181no—
375C22—181no—
376C22C3⋊D3×D1211yescharacteristic
377C4—62no—
378C4—121no—
379C4—181no—
380C4—542no—
381C4—1081no—
382C4D3×C4⋊(C3⋊D3)11yescharacteristic
383C4#8903911yescharacteristic
384C3—24no—
385C3#1059095214yescomplemented
386C3#1059250811yescharacteristic, complemented
387C2—182no—
388C2C3⋊D3×Dic1211yescharacteristic
389C2#1172334011yescharacteristic
390C2#1172354211yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#451
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#88853C2semidirect
4. C2×#88799C2semidirect
6. C32×#816C2semidirect
7. #11722041C2semidirect
38. C3×#816D3semidirect
98. C2×C3×C3⋊D3Q16semidirect
119. #816C3⋊D3semidirect
211. C2×C3⋊D3Dic12semidirect
240. C33#451semidirect
341. C32#4739semidirect
342. C32#4809semidirect
385. C3#10590952semidirect
386. C3#10592508semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3⋊D3×Dic62
C2×#888532
#117234312
C2×#887992
C3×#636472
C32×#8162
#117220412
#105925083
#105909523